Concerning the mathematical analysis, the Cauchy problems corresponding to () and (), each endowed with (), have been studied in [1]}, [2]}, [3]}, [4]} and in
[5]}, [6]}, [7]}, [8]}, [9]}, respectively. We also refer the interested reader to [10]} (see also [11]}) for the connection between the above DI systems and the two-phase Navier–Stokes free boundary problem.
